One of the most disconcerting, but profoundly beautiful pieces of Mathematics are Kurt Gödel’s incompleteness theorems, which he proved in 1931 and which show that “no consistent system of axioms […] is capable of proving all truths about the relations of the natural numbers […]. For any such system, there will always be statements about the natural numbers that are true, but that are unprovable within the system [… and] that such a system cannot demonstrate its own consistency.”
Here axioms are a system’s premises or starting points that are taken for granted (as self-evident or as expressing a property of the entities the system refers to), and are not provable within that system, and theorems are statements derived from these axioms. Gödel’s theorems therefore say that no matter how complex a system of consistent axioms (i.e., axioms that cannot lead both to a theorem and its negation), the set of all possible theorems derived from it will not include all true statements about natural numbers. In other words, that there exists an arithmetic statement that is true but not provable within that system of axioms (i.e., not derivable from them). Gödel achieves this using an ingenious device - the so-called “Gödel sentence” - which in essence claims that it (the Gödel sentence) cannot be proved within a given, consistent axiomatic system.
If this theorem (the Gödel sentence) could be proved using a system’s axioms, then the system would contain a theorem that contradicts itself (i.e., the theorem stating that it cannot be proved would be proved). The system would therefore be inconsistent. However, since the axiomatic system is consistent, the theorem cannot be proved within it. The system’s consistency renders the theorem both true and outside the system. The system is therefore incomplete (not containing the true Gödel sentence) and provability-within-a-system-of-axioms is not the same as truth. This is the gist of the first of the two theorems, put in as plain language as I could manage.1
If you are still reading this, I guess you may be wondering “when do we get to the Church teaching bit?” and I apologize for the unusually lengthy preamble and for passing through the following de-tour, before attempting a bringing together of the strands I set out in the title. That detour regards the Theory of Everything (that I name-dropped in an earlier post) and the death-blow it was dealt by Gödel’s theorems. The Theory of Everything is “a putative theory of theoretical physics that fully explains and links together all known physical phenomena, and predicts the outcome of any experiment that could be carried out in principle.” While such a theory does not exist, for a long time it has been the goal that science has been striving for and that it believed to be progressing towards. One day it would arrive at a level of understanding of the universe that would allow it to predict any event and to do so using a single, unified theory.
Without going into to the varied arguments for the impossibility of such a theory, let me just quote Stephen Hawking:
“What is the relation between Gödel’s theorem, and whether we can formulate the theory of the universe, in terms of a finite number of principles. One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted. One example might be the Goldbach conjecture. Given an even number of wood blocks, can you always divide them into two piles, each of which can not be arranged in a rectangle. That is, it contains a prime number of blocks.So, what does all of this have to do with Church teaching or with the Holy Spirit? Well, if you look at any system of reasoning, where some statements are derived from others and where validity can be determined by comparing a statement with the system’s premises according to its rules of reasoning (themselves being premises), then such a system can be seen as having an underlying mathematical model, which thanks to Gödel is now forever revealed as incomplete. The Church’s teaching, as a set of premises (e.g., dogmas, Scripture, etc.) and statements derived from them, is therefore also subject to Gödel’s catch and, even from the perspective of logic alone, incapable of claiming to contain all truth.
[…]
Some people will be very disappointed if there is not an ultimate theory, that can be formulated as a finite number of principles. I used to belong to that camp, but I have changed my mind. I’m now glad that our search for understanding will never come to an end, and that we will always have the challenge of new discovery. Without it, we would stagnate.”
Before you shout “Blasphemy!”,2 let me argue that this is neither negative nor new. The Church aims to proclaim the Good News of God’s love that Jesus brought both by his own teaching and - completely - in his own person. When Jesus says during the Last Supper: “I am in the Father and the Father is in me” (John 14:11) and that “[I have] been with you for so long a time and you still do not know me”, he is telling us that he - Jesus the person - is the message and that he cannot be reduced only to the teachings he explicitly shared with his disciples during those three short years of public ministry.
In fact, he proceeds to tell those assembled in the Upper Room that: “The Advocate, the holy Spirit that the Father will send in my name—he will teach you everything and remind you of all that [I] told you.” (John 14:26). During his last address to his followers, Jesus emphasizes the fact that the Church will, to use Hawking’s words “always have the challenge of new discovery,” thanks to the Holy Spirit, who will supply it with a continuous stream of inputs.3 Essentially, the Church can claim to have the Truth insofar as it is the Mystical Body of Jesus, who is its head, who is one with the Father and the Holy Spirit, who is God and who therefore is the fullness of Truth. As far as its explicit, finite set of teachings is concerned, it is subject to incompleteness. This is so not only because of the underlying limitations of any system that employs premises and statements derived from them (as Gödel’s theorems prove), but also because God, who is infinite, always-greater, cannot be encapsulated in a set of human-readable rules and statements. If we thought otherwise and viewed the Church’s teaching (qua teaching) as complete and comprehensive, “we would stagnate” (again borrowing Prof. Hawking’s words).
I believe the above is highly consistent with how Benedict XVI presented the aims of the Year of Faith that is currently in progress. Instead of calling the members of the Church to swat up on its rules and regulations, he invited them to “an encounter with a Person,” a “friendship with the Son of God.” This does not mean that knowledge of the understanding that the Church has gained since Jesus walked the Earth is not valuable (it is!), but that the Christian faith is “no theory.” To conclude, Benedict sums the centrality of the person of Jesus up as follows:
“The joy of love, the answer to the drama of suffering and pain, the power of forgiveness in the face of an offence received and the victory of life over the emptiness of death: all this finds fulfilment in the mystery of his Incarnation, in his becoming man, in his sharing our human weakness so as to transform it by the power of his resurrection.”
I would like to thank my überbestie, PM, for the sanity check, his Nihil Obstat and Transferitur.
1 For those of you who are mathematically inclined, the Wikipedia page on the incompleteness theorems both contains a sketch of the proof (including his beautiful arithmetization syntax, which allows for the Gödel sentence's expression in arithmetic axiomatic systems) and points to more in-depth material. It also addresses how even adding the Gödel sentence to a set of axioms (i.e., making the Gödel sentence an axiom of a system) fails to defeat it :).
2 And proceed to purchase a small packet of gravel from Harry the Haggler.
3 E.g., see also the Second Vatican Council’s Dei Verbum saying: “Sacred tradition and Sacred Scripture form one sacred deposit of the word of God, committed to the Church.” and “This tradition […] comes from the Apostles [and is] develop[ed] in the Church with the help of the Holy Spirit.”
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